Z 2 -equivariant standard bases for submodules associated with Z 2 -equivariant singularities

Abstract

Let x = ( x 1 ,..., x n ) ∈ R n and λ ∈ R. A smooth map f ( x ,λ) is called Z 2 -equivariant ( Z 2 -invariant) if f (− x , λ) = − f ( x ,λ) ( f (− x , λ) = f ( x ,λ)). Consider the local solutions of a Z 2 -equivariant map f ( x ,λ) = 0 around a solution, say f ( x 0 ,λ 0 ), as the parameters smoothly vary. The solution set may experience a surprising behavior like observing changes in the number of solutions. Each of such problems/changes is called a singularity/bifurcation. Since our analysis is about local solutions, we call any two smooth map f ( x ,λ) and g ( x ,λ) as germ-equivalent when they are identical on a neighborhood of ( x 0 ,λ 0 ) = (0,0). Each germ equivalent class is referred to a smooth germ. The space of all smooth Z 2 -equivariant germs is denoted by [EQUATION] and space of all smooth Z 2 -invariant germs is denoted by [EQUATION] x ,λ ( Z 2 ). The space [EQUATION] is a module over the ring of Z 2 -invariant germs [EQUATION] x ,λ ( Z 2 ); see [3, 2, 7] for more information and the origins of our notations.